4 XII TOPOLOGY AND TOPOLOG1CAL ALGEBRA The equivalence of (a) and (b) follows from the criterion of continuity (3.11.4(b)), and the equivalence of (b) and (c) from the definition of a con- tinuous function. Remark (12.2.2) Let (Ua)a€, be an open covering (3.16) of a topological space E. For a set G c E to be open in E it is necessary and sufficient that each of the sets G n Ua should be open in the subspace Ua. This follows immediately from the axioms (Oj) and (On) and the relation G = (J (G n Ua). Taking complements, ael it follows that a set F c E is closed m E if and only if each of the sets F n Ua is closed in the subspace Ua. (12.2.3) Let L be a subset of a topological space E. Then the following proper- ties are equivalent: (a) For each xelu there exists a neighborhood V of x in E such that L n V is closed in V; (b) L is an open subset of the subspace L (the closure ofL in E); (c) L is the intersection of an open subset and a closed subset of'E. It is clear that (b) implies (c), L being the intersection of L with an open subset of E. Equally clearly, (c) implies (a). Let us show that (a) implies (b). For each x e L, we have V n L = V n L, because V n L is closed in V; this shows that in the subspace L the point x is an interior point of L, and therefore L is open in L. When L satisfies the equivalent conditions of (12.2.3), it is said to be a locally closed subset of E. (12.2.4) A procedure for constructing topological spaces which is frequently used is that of "patching together" a family of topological spaces (EA)AeL in such a way that, in the topological space E so obtained, the EA are identified with open sets of E. Since pairs of these sets may very well intersect, this identi- fication requires that, for each pair of indices (1, u), we are given a homeomor- phism of an open subset of EA onto an open subset of EM. To be precise, suppose that for each pair (1, /^) e L x L we are given: (1) an open subset AA/[t of EA; (2) a homeomorphism h^: AA/i -* A^, satisfying the following conditions:efinitions and results of (3.19) relating to